Stabilization of Vibrating Beam with a Tip Mass Controlled by Combined Feedback Forces Original Research Article

A flexible structure consisting of a Euler–Bernoulli beam with a tip mass is considered. To stabilize this system we use a boundary control laws: −uxxx(1, t) + mutt(1, t) = −αut(1, t) + βuxxxt(1, t) and uxx(1, t) = −γuxt(1, t). A sensitivity asymptotic analysis of the systemʹs eigenvalues and eigenfunctions is set up. We prove that all of the generalized eigenfunctions of (2.9) form a Riesz basis of H. By a new method, we prove that the operator A generates a C0 contraction semigroup T(t), t ≥ 0. Furthermore T(t), t ≥ 0, is uniformly exponentially stable and the optimal exponential decay rate can be obtained from the spectrum of the system.